the disciple of Philolaos, and Aristoxenos mentioned him along with Philolaos as having taught the last of the Pythagoreans, the men with whom he himself was acquainted. He therefore belongs to the beginning of the fourth century B.C., by which time the Pythagorean system was fully developed, and he was no eccentric enthusiast, but one of the foremost men in the school.[1] We are told of him, then, that he used to give the number of all sorts of things, such as horses and men, and that he demonstrated these by arranging pebbles in a certain way. Moreover, Aristotle compares his procedure to that of those who bring numbers into figures (σχήματα) like the triangle and the square.[2]
Now these statements, and especially the remark of Aristotle last quoted, seem to imply the existence at this date, and earlier, of a numerical symbolism quite distinct from the alphabetical notation on the one hand and from the Euclidean representation of numbers by lines on the other. The former was inconvenient for arithmetical purposes, because the zero was not yet invented.[3] The representation of numbers by lines was adopted to avoid
- ↑ Apart from the story in Iamblichos (V. Pyth. 148) that Eurytos heard the voice of Philolaos from the grave after he had been many years dead it is to be noticed that he is mentioned after him in the statement of Aristoxenos referred to (Diog. viii. 46; R. P. 62).
- ↑ Arist. Met. N, 5. 1092 b 8 (R. P. 76 a). Aristotle does not quote the authority of Archytas here, but the source of his statement is made quite clear by Theophr. Met. p. vi. a 19 (Usener), τοῦτο γὰρ (sc. τὸ μὴ μέχρι του προελθόντα παύεσθαι) τελέου καὶ φρονοῦντος, ὅπερ Ἀρχύτας ποτ' ἔφη ποιεῖν Εὔρυτον διατιθέντα τινὰς ψήφους· λέγειν γὰρ ὡς ὅδε μὲν ἀνθρώπου ὁ ἀριθμός, ὅδε δὲ ἵππου, ὅδε δ' ἄλλου τινὸς τυγχάνει.
- ↑ The notation used in Greek arithmetical treatises must have originated at a date and in a region where the Vau and the Koppa were still recognised as letters of the alphabet and retained their original position in it. That points to a Dorian state (Taras or Syracuse?), and to a date not later than the early fourth century B.C. The so-called Arabic figures are usually credited to the Indians, but M. Carra de Vaux has shown (Scientia, xxi. pp. 273 sqq.) that this idea (which only makes its appearance in the tenth century A.D.) is due to a confusion between the Arabic hindi, "Indian," and hindasi, "arithmetical." He comes to the conclusion that the "Arabic" numerals were invented by the Neopythagoreans, and brought by the Neoplatonists to Persia, whence they reached the Indians and later the Arabs. The zero, on which the value of the whole system depends, appears to be the initial letter of οὐδέν.
